Discrete Inf-sup Condition Research Articles

A stabilized finite volume method based on the rotational pressure correction projection for the time-dependent incompressible MHD equations

This paper presents an efficient numerical scheme for solving the time-dependent incompressible Magnetohydrodynamics (MHD) equations based on the rotational pressure correction projection method within the finite volume framework. Utilizing the lowest equal-order mixed finite element pair (P1−P1−P1) to approximate the velocity, magnetic and pressure fields, our numerical scheme satisfies the discrete inf-sup condition by the pressure projection stabilization. To tackle the coupling inherent in the time-dependent incompressible MHD equations,the rotational pressure correction projection method is introduced to split the original problem into several linear subproblems. The unconditional stability of numerical schemes are provided,optimal error estimates in both L2 and H1-norms of numerical solutions are also presented.Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical schemes.

An extended nonconforming finite element method for the coupled Darcy–Stokes problem

An extended nonconforming finite element method for solving the coupled Darcy–Stokes problem with straight or curved interfaces is proposed and analyzed. The approach applies the same Crouzeix–Raviart discretization in both regions. By introducing some stabilization terms, the discrete inf-sup condition and optimal a priori estimate are derived. In the end, some numerical experiments are presented to demonstrate the theoretical results.

Fully decoupled, linearized and stabilized finite volume method for the time-dependent incompressible MHD equations

In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (P1-P1-P1) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.

New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

A low-degree strictly conservative finite element method for incompressible flows on general triangulations

In this study, a new P 2 -P 1 finite element pair is proposed for incompressible fluid. For this pair, the discrete inf-sup condition and the discrete Korn’s inequality hold for general triangulations. It yields strictly conservative velocity approximations when applied to models of incompressible flows. The convergence rate of the scheme can only be proved to be of suboptimal 𝒪(h) order, though, based on the property of strict conservation, the robust capacity of the pair for incompressible flows is verified theoretically and numerically.

Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh.

In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby bypassing the obstacle of standard finite element discretizations on arbitrary mesh. The velocity and pressure are approximated by the practical significative lowest equal-order virtual element space pair which does not satisfy the discrete - condition. Combined with the penalty method, the error estimation is proved rigorously. Numerical results on the 3D polygonal mesh illustrate the theoretical results and effectiveness of the proposed method.

Stability and convergence of spatial discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics equations

In this paper, the stability and convergence results of finite volume method for the time-dependent incompressible magnetohydrodynamics equations are developed based on triangulation. The linear polynomials are used to approximate the velocity, magnetic fields and pressure. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection method is employed. Firstly, two self-adjoint linear mappings are introduced to establish the equivalences of temporal terms and bilinear terms in finite element method and finite volume method. Secondly, the existence and uniqueness of numerical solutions are presented by using the Oseen iteration and mathematical induction method. Thirdly, H2-stability and optimal error estimates are provided by introducing the Galerkin projection of generalized bilinear terms in finite volume method, using the energy method and constructing the corresponding dual problem. The main theoretical findings of this paper are as follows:‖(uh(t),Hh(t))‖02+∫0t(min⁡{ν,γσ−1}‖(uh,Hh)‖12+G(ph,ph))ds≤C,min⁡{ν,γσ−1}‖(uh(t),Hh(t))‖12+G(ph(t),ph(t))+∫0t‖(uht,Hht)‖02ds≤C,‖∇(uh,Hh)‖02+∫0tmin⁡{ν,γσ−1}‖(A1huh,A2hHh)‖02ds≤C,τ12(t)‖(u(t)−uh(t),H(t)−Hh(t))‖0+h(‖(u(t)−uh(t),H(t)−Hh(t))‖1+τ12(t)‖p(t)−ph(t)‖0)≤Ch2, where τ(t)=min⁡{1,t}, the letters ν,σ are the physical parameters, C and γ are two positive constants independent of the mesh size h, (u,H,p) and (uh,Hh,ph) are the solutions of problems (2.1) and (4.1). Finally, some numerical results are presented to verify the established theoretical findings and show the performances of the considered numerical method.

Fully discrete best-approximation-type estimates inL∞ (I;L2(Ω)d) for finite element discretizations of the transient Stokes equations

AbstractIn this article, we obtain an optimal best-approximation-type result for fully discrete approximations of the transient Stokes problem. For the time discretization, we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions.

Difference finite element method for the 3D steady Stokes equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution (uh,ph) of the 3D steady Stokes equations in the direction of (x,y,z) into a series of the finite element solution (uhk,phk) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair (P1b,P1b,P1)×P1, where the 2D finite element pair (P1b,P1b)×P1 satisfies the discrete inf-sup condition in a 2D domain ω. Here we design the weak formulation of the DFE method based on the 3D finite element pair ((P1b,P1b,P1)×P1)×(P1×P0) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution (uh,ph)=(∑k=0l3uhkϕk(z),∑k=1l3phkψk(z)) and deduce the first order convergence of the DFE solution (uh,ph) with respect to the exact solution (u,p) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method.

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations.

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair . The method consists of transmitting the finite element solution of the 3D steady Navier–Stokes equations into the finite element solution pairs based on the finite element space pair of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair satisfies the discrete inf-sup condition in a 3D domain . Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to of the FE solution to the exact solution of the 3D steady Navier–Stokes equations in the norm. Finally, we also give the convergence order with respect to of the FE velocity to the exact velocity u of the 3D steady Navier–Stokes equations in the norm.

Adaptive virtual element methods with equilibrated fluxes

We present an hp-adaptive virtual element method (VEM) based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems. We introduce a reliable and efficient a posteriori error estimator, which is computed by solving an auxiliary global mixed problem. We show that the mixed VEM satisfies a discrete inf-sup condition with inf-sup constant independent of the discretization parameters. Furthermore, we construct a stabilization for the mixed VEM with explicit bounds in terms of the local degree of accuracy of the method. The theoretical results are supported by several numerical experiments, including a comparison with the residual a posteriori error estimator. The numerics exhibit the p-robustness of the proposed error estimator. In addition, we provide a first step towards the localized flux reconstruction in the virtual element framework, which leads to an additional reliable a posteriori error estimator that is computed by solving local (cheap-to-solve and parallelizable) mixed problems. We provide theoretical and numerical evidence that the proposed local error estimator suffers from a lack of efficiency.

Stability Analysis of Polytopic Discontinuous Galerkin Approximations of the Stokes Problem with Applications to Fluid–Structure Interaction Problems

We present a stability analysis of the Discontinuous Galerkin method on polygonal and polyhedral meshes (PolyDG) for the Stokes problem. In particular, we analyze the discrete inf-sup condition for different choices of the polynomial approximation order of the velocity and pressure approximation spaces. To this aim, we employ a generalized inf-sup condition with a pressure stabilization term. We also prove a priori hp-version error estimates in suitable norms. We numerically check the behaviour of the inf-sup constant and the order of convergence with respect to the mesh configuration, the mesh-size, and the polynomial degree. Finally, as a relevant application of our analysis, we consider the PolyDG approximation for a 2D fluid–structure interaction problem and we numerically explore the stability properties of the method.

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman–Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška–Brezzi’s theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimala priorierror estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method.

Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations

Abstract Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete inf-sup condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization

In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$-regularization and with a sparse optimal control approach.

A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem

Combining the mixed discontinuous Galerkin method for the Darcy flow and the interior penalty discontinuous Galerkin methods for the Stokes problem, a locally conservative discrete scheme is proposed for numerically solving the coupled Stokes and Darcy problem. We prove the well-posedness of the solution of the proposed numerical scheme by boundedness, K-ellipticity and a discrete inf-sup condition. A priori error estimates, in proper norms are derived, and to verify the theoretical analysis, some numerical experiments are given.

A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem

We propose and analyze a stabilized virtual element method for the Stokes problem on polytopal meshes. We employ the $$C^0$$ continuous arbitrary “equal-order” virtual element pairs to approximate both velocity and pressure, and develop a projection-based stabilization term to circumvent the discrete inf-sup condition, then we obtain the corresponding error estimates. The presented method involves neither the projection of the second derivative nor additional coupling terms, and it is parameter-free. In particularly, for the lowest-order case on triangular (tetrahedral) meshes the stabilized method introduced by Bochev et al. (SIAM J. Numer. Anal. 44: 82–101, 2006) is a special case of our method up to an approximation of the load term. Furthermore, numerical results are shown to confirm the theoretical predictions.